Kevin Hart was the victim in an extortion attempt to the tune of millions of dollars, and we're told the FBI is on the case and has a suspect.

Law enforcement sources tell us ... an anonymous person contacted Kevin's camp and provided the video TMZ described Saturday ... showing Kevin and a woman in a sexually provocative situation. The video then cuts to a bedroom where there appears to be sexual activity on a bed, although you don't see who is in the bed.

We're told the person demanded a multi-million dollar amount in return for keeping the video private. There were actually several demands, and at one point the amount hit 8 figures.

Our sources say The FBI knows who the woman is who interacted with Kevin, and they either believe she is the one who made the money demand ... or someone who got hold of the iPhone that recorded the encounter made the demand. Either way, FBI agents think they know the identity of the culprit.

Sources close to Kevin will not say if he's the one in the bedroom, although they do say the woman who is on the various videos has had sexual relations with another celebrity -- a TV star -- and they believe he might be the one in the bedroom videos.

Our Kevin sources also say the videos were recorded on an iPhone that was propped against an object in the room and such recording without consent is illegal.